Cagniard--de Hoop path perturbations with applications to non-geometric wave arrivals

Non-geometric wave arrivals are often important in seismology and elastic wave studies related to the non-destructive evaluation of structures. In particular tunnelling signals caused by significant differences in the material parameters, and wavespeeds at interfaces, generate large responses that may often be dominant. This is common in elastic wave propagation, for instance, when a source is close to the interface of a faster medium with a slower medium, the response in the slower medium is dominated by a signal that has tunnelled through the faster medium. Other instances of tunnelling occur when a compressional source is close to a free surface. In this case the compressional to shear wave conversion at the surface, and the mismatch between compressional and shear wavespeeds, leads to a sharp non-geometric shear wave arrival. Equally, thin high velocity layers demonstrate tunnelling effects that are perturbations of the response brought about by a source in a surrounding slower medium. In the above close refers to the viewpoint of an observer some distance away. In all of the instances there is a common feature, namely, each problem contains a ratio of length scales, x/h, with h either the source depth or layer thickness and x the observer distance; this ratio of length scales characterises the non-geometric responses. Typically, the non-geometric response arises when the current problem is a perturbation away from one where the associated arrival has a direct geometric interpretation. Such problems are ideally suited to analysis by the Cagniard-de Hoop technique. Each tunnelling response is identified as a perturbation away from an exact solution; this leads to highly accurate and relatively simple explicit asymptotic solutions. The perturbation scheme is demonstrated here via the solution of two problems: a compressional source beneath a fluid/solid interface and beneath a thin high velocity layer. The first problem has separate non-geometric responses due to both the material mismatch and the wave conversion at the interface. The thin high velocity layer perturbs the field generated by a compressional source in a slower surrounding medium. In both cases the non-geometric arrivals are analysed in detail.

Co-authored with Duncan Williams.

To appear J. Engng. Maths

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